3.4 Filters

Definition 3.4.1 (Filters).label Let $X$ be a set then $\scr{F}\suf\cali{P}(X)$ is a filter on $X$ if it satisfies the following:

1. (F1)

   For each $F\in\scr{F}$ and $F\suf E\suf X$, $E\in\scr{F}$

2. (F2)

   For each $I$ finite index set and $\curl{E_i}_{i\in I}\suf\scr{F}$, $\caps{}{i\in I}E_{i}\in\scr{F}$

3. (F3)

   $\emp\nin\scr{F}$

(F2) is equivalent to the following axioms:

1. (F2a)

   For each $E,F\in\scr{F}$, $E\cap F\in \scr{F}$

2. (F2b)

   $X\in\scr{F}$

Let $\scr{F},\scr{F}'$ be filters on $X$. If $\scr{F}\suf \scr{F}'$ then $\scr{F}'$ is (strictly) finer than $\scr{F}$ and $\scr{F}$ is (strictly) coarser than $\scr{F}'$ (if in addition $\scr{F}\neq\scr{F}'$). The set of all filters on $X$ is ordered by this relation.

Definition 3.4.2 (Generator and Subbase).label Let $X$ be a set, $\scr{O}\suf\cali{P}(X)$ and $\scr{F}$ a filter on $X$. We say $\scr{F}$ is the filter generated by $\scr{O}$ if $\scr{O}\suf\scr{F}$ is coarser than any filter $\scr{F}'$ that contains $\scr{O}$, and $\scr{O}$ is said to be a subbase of $\scr{F}$.

There exists a filter $\scr{F}$ containing $\scr{O}\suf \cali{P}(X)$ if and only if $\fall \curl{E_i}_{i\in I}\suf \scr{O}$ such that $I$ is finite, $\caps{}{i\in I}E_{i}\neq\emp$. In which case the filter generated by $\scr{O}$ has the form

Definition 3.4.3 (Filter Base).label Let $X$ be a set, $\scr{F}$ a filter and $\scr{B}\suf \scr{F}$, then $\scr{B}$ is a filter base for $\scr{F}$ if